Definition

1. Noun:
   • A branch of non-Euclidean geometry in which space is modeled as having properties analogous to the surface of a sphere. In this geometry, there are no parallel lines, and the sum of the angles in a triangle is always greater than 180 degrees.

Usage

• Elliptic geometry provides a mathematical framework for describing curved spaces, such as the surface of a sphere.
• The study of elliptic geometry is fundamental to understanding the geometry of the universe in certain cosmological models.
• In elliptic geometry, the concept of a "straight line" is replaced by the concept of a geodesic, like a great circle on a sphere.

Examples

• The mathematician explored the properties of triangles in elliptic geometry.
• Elliptic geometry challenges the Euclidean postulate that through a point not on a given line, exactly one line can be drawn parallel to the given line.
• Understanding elliptic geometry is essential for certain applications in astronomy and general relativity.

Advanced Usage

• Riemannian geometry is a broad generalization that includes elliptic geometry as a special case.
• The geometry of a sphere is a simple, intuitive model for understanding the principles of elliptic geometry.

Variants and Related Words

• Elliptical (adjective): Pertaining to or having the form of an ellipse. (Note: While related, "elliptical" in common usage often refers to the shape of an oval, whereas "elliptic" in this context refers to a specific type of geometry).
• Riemannian geometry (noun): A more general form of non-Euclidean geometry developed by Bernhard Riemann, which includes elliptic geometry.
• Spherical geometry (noun): Often used synonymously with elliptic geometry, especially in its two-dimensional form on the surface of a sphere.

Synonyms

• Riemannian geometry (in a specific sense)
• Spherical geometry

Related Concepts (Not Phrasal Verbs or Idioms)

• Non-Euclidean geometry: The overarching category of geometries that do not obey all of Euclid's postulates, which includes elliptic geometry.
• Great circle: The analogue of a "straight line" in the spherical model of elliptic geometry; it is the intersection of a sphere with a plane passing through its center.
• Gaussian curvature: A measure of the intrinsic curvature of a surface; elliptic geometry is characterized by constant positive curvature.

Noun

1. (mathematics) a non-Euclidean geometry that regards space as like a sphere and a line as like a great circle
   • Bernhard Riemann pioneered elliptic geometry